Stochastic structural optimisation with quadratic loss functions

Problems from plastic analysis and optimal plastic design are based on the convex, linear or linearised yield/strength condition and the linear equilibrium equation for the stress (state) vector. In practice one has to take into account stochastic variations of several model parameters. Hence, in order to get robust optimal decisions, the structural optimisation problem with random parameters must be replaced by an appropriate deterministic substitute problem. A direct approach is proposed based on the primary costs (weight, volume, costs of construction, costs for missing carrying capacity, etc.) and the recourse costs (e.g. costs for repair, compensation for weakness within the structure, damage, failure, etc.). Based on the mechanical survival conditions of plasticity theory, a quadratic error/loss criterion is developed. The minimum recourse costs can be determined then by solving an optimisation problem having a quadratic objective function and linear constraints. For each vector image of model parameters and each design vector x, one obtains then an explicit representation of the “best” internal load distribution image. Moreover, also the expected recourse costs can be determined explicitly. Consequently, an explicit stochastic nonlinear program results for finding a robust optimal design image, a maximal load factor. The analytical properties and possible solution procedures are discussed.